How do you find local maximum value of f using the first and second derivative tests: #2x^33x^236x3#?
There is a local maximum of
 Find the critical values of the function. These are where the local maximum could exist.
To find this function's derivative, use the power rule.
These are the function's two critical values. These are where the local maxima or minima could exist.
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 Use the first derivative test to see if the function switches from increasing to decreasing or decreasing to increasing around the critical values.
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 Examine the concavity at each critical value through the second derivative (the second derivative test).
First, find the second derivative by differentiating the first derivative.
The process for the second derivative test is as follows:
Note that we should get the same results that the first derivative test gave usthis is just another way of reaching the same conclusion.
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If you're able, always consult a graph of the function to check your answer:
graph{2x^33x^236x3 [5, 7, 106.8, 107]}
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To find the local maximum value of ( f(x) = 2x^3  3x^2  36x  3 ) using the first and second derivative tests:

Find the first derivative of ( f(x) ): [ f'(x) = 6x^2  6x  36 ]

Find the critical points by setting the derivative equal to zero and solving for ( x ): [ 6x^2  6x  36 = 0 ] [ x^2  x  6 = 0 ] [ (x  3)(x + 2) = 0 ] [ x = 3, 2 ]

Determine the nature of the critical points:
 At ( x = 3 ):
 ( f'(3) = 6(3)^2  6(3)  36 = 0 )
 Use the second derivative test: ( f''(x) = 12x  6 )
 ( f''(3) = 12(3)  6 = 30 > 0 ), so it's a local minimum.
 At ( x = 2 ):
 ( f'(2) = 6(2)^2  6(2)  36 = 0 )
 Use the second derivative test: ( f''(x) = 12x  6 )
 ( f''(2) = 12(2)  6 = 30 < 0 ), so it's a local maximum.
 At ( x = 3 ):

Calculate the value of ( f(x) ) at the local maximum point: [ f(2) = 2(2)^3  3(2)^2  36(2)  3 ] [ f(2) = 47 ]
Therefore, the local maximum value of ( f(x) ) occurs at ( x = 2 ), and its value is ( 47 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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